In the US, it's done on digital recorders, and the audio is kept for (IIRC) 45 days before it's overwritten or copied to an optical disc due to an investigation. Everything is searchable by date, time, and position worked, so we can drill down to the second when searching for something.

Internationally? Sometimes I think this site may be the only record of some locations.

Unfortunately, the UK has a law that prohibits listening to and/or streaming of any ATC communications. It's for this reason that there are no Liveatc feeds from the UK. And it's also the same reason there are no Liveatc feeds from Germany or Italy.